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\begin{document}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
{\Large\bf Towards the cone conjecture\\
for hyperk\"ahler manifolds}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\\[14mm]

{\small Misha Verbitsky } 
\\[20mm]

{\tiny\bf 
Simons Center \\
Quiver Varieties Program Seminar\\
November 8, 2013
}
\end{center}

\newpage

{\bf \blue The K\"ahler cone and its faces}

\def\Teich{\operatorname{Teich}}
\newcommand{\Comp}{\operatorname{Comp}}
\newcommand{\Kah}{\operatorname{\sf Kah}}
\newcommand{\Bir}{\operatorname{\sf Bir}}
\newcommand{\Pos}{\operatorname{\sf Pos}}


The work presented here (on the slides 13-15) {\bf \red is 
done in collaboration with Ekaterina Amerik.}


\definition
Let $M$ be a compact, K\"ahler manifold,
$\Kah\subset H^{1,1}(M,\R)$ is K\"ahler cone,
and $\overline\Kah$ its closure in $H^{1,1}(M,\R)$,
called {\bf \blue the nef cone}. A {\bf\blue face} of a 
K\"ahler cone is an intersection
of the boundary of $\overline\Kah$ and a hyperplane $V\subset H^{1,1}(M,\R)$
which has a non-empry interior.

\conjecture {\bf \blue (Morrison-Kawamata cone conjecture)} \\
Let $M$ be a Calabi-Yau manifold. Then 
the group $\Aut(M)$ of biholomorphic automorphisms of $M$
acts on the set of faces of $\Kah$ {\bf \red with finite number of orbits.}


\newpage

{\bf \blue Birational K\"ahler cone}


\remark
Define {\bf \blue pseudo-isomorphism} $M\arrow M'$ as a
birational map which is an isomorphism outside of codimension
$\geq 2$ subsets of $M, M'$.

\remark {\bf \purple For any pseudo-isomorphic manifolds $M,M'$,
one has $H^2(M)=H^2(M')$.}

\definition
{\bf \blue Movable K\"ahler cone}, also known as {\bf \blue birational
K\"ahler cone} and {\bf \blue birational nef cone}
is a closure of a union of $\Kah(M')$ for all
$M'$ pseudo-isomorphic to $M$.

\conjecture {\bf \blue(Morrison-Kawamata birational cone conjecture)}\\
Let $M$ be a Calabi-Yau manifold. Then 
the group $\Bir(M)$ if birational automorphisms of $M$
{\bf \red acts on the set of faces of the movable cone with finite 
number of orbits.}


\newpage

{\bf \blue $(-2)$-classes on a K3 surface }

\claim {\bf \blue (Hodge index theorem)} \\
Let $M$ be a K\"ahler surface. {\bf \red Then the form 
$\eta \arrow \int_M\eta\wedge\eta$ has signature
$(+, -, -, ...)$ on $H^{1,1}(M,\R)$.}

\definition
{\bf \blue Positive cone} $\Pos(M)$ on a K\"ahler surface
is the one of the two components of 
\[ \{v\in H^{1,1}(M,\R)\ \ |\ \  \int_M\eta\wedge\eta>0\}
\]
which contains a K\"ahler form.

\definition
A cohomology class $\eta\in H^2(M,\Z)$ on a K3 surface
is called {\bf \blue $(-2)$-class} if $\int_M\eta\wedge\eta=-2$.

\remark
Let $M$ be a K3 surface, and $\eta\in H^{1,1}(M,\Z)$ a $(-2)$-class.
{\bf \purple Then either $\eta$ or $-\eta$ is effective.} Indeed,
$\chi(\eta) = 2+\frac{\eta^2}{2}=1$ by Riemann-Roch.

\newpage

{\bf \blue K\"ahler cone for a K3 surface}

\theorem
Let $M$ be a K3 surface, and $S$ the set of all effective $(-2)$-classes.
{\bf \red Then $\Kah(M)$ is the set of all $v\in \Pos(M)$
such that $\langle v, s\rangle >0$ for all $s\in S$.}



\definition
{\bf \blue A Weyl chamber} on a K3 surface is a connected component
of $\Pos(M)\backslash S^\bot$, where $S^{\bot}$ is a union of
all planes $s^\bot$ for all (-2)-classes $s\in S$. 
{\bf \blue The reflection group}
of a K3 surface is a group $W$ generated by reflections with
respect to all $s\in S$.

\remark Clearly, a Weyl chamber is a fundamental domain of $W$,
and $W$ acts transitively on the set of all Weyl chambers.
Moreover, {\bf \purple
the K\"ahler cone of $M$ is one of its Weyl chambers.}

\newpage

{\bf \blue Cone conjecture for a K3 surface}

\theorem
Let $M$ be a K3 surface. {\bf \red Then $\Aut(M)$ is the group of all 
isometries of $H^{1,1}(M,\Z)$ preserving the K\"ahler chamber.}

\corollary 
{\bf \red Morrison-Kawamata cone conjecture holds for a K3 surface.}

{\bf \green Proof. Step 1:}  A group $\Gamma$ of isometries
of a lattice $\Lambda$ acts with finitely many orbits on the
set $\{l \in \Lambda\ \ |\ \ l^2 = x\}$
for any given $x$ (see Kneser, {\em Quadratische Formen},
Satz 30.2). {\bf \purple Therefore, $\Gamma$ acts with finitely many
orbits on the set of $(-2)$-vectors in $\Lambda$.}

{\bf \green Step 2:}
For each pair of faces $F, F'$ of a K\"ahler
cone and $w\in O(\Lambda)$ mapping $F$ to $F'$, $w$ maps $\Kah$ to itself
or to an adjoint Weyl chamber $K'$. Then $K'=r(K)$, where
$r$ is the reflection fixing $F'$.
In the first case, $w\in \Aut(M)$.
In the second case, $rw$ maps $F$ to $F'$ and maps $\Kah$ to itself,
hence $rw\in \Aut(M)$.
\endproof


\newpage

{\bf \blue Hyperk\"ahler manifolds}

\definition
A {\bf \blue hyperk\"ahler structure} on a manifold $M$
is a Riemannian structure $g$ and a triple of complex
structures $I,J,K$, satisfying quaternionic relations
$I\circ J = - J \circ I =K$, such that $g$ is K\"ahler
for $I,J,K$.

\remark A hyperk\"ahler manifold {\bf \purple has three symplectic forms\\
$\omega_I:=  g(I\cdot, \cdot)$, $\omega_J:=  g(J\cdot, \cdot)$,
$\omega_K:=  g(K\cdot, \cdot)$.}


{\bf\green REMARK:} This is equivalent to $\nabla I=\nabla J = \nabla K=0$:
the parallel translation along the connection preserves $I, J,K$.


{\bf\green DEFINITION:} 
Let $M$ be a Riemannian manifold, $x\in M$ a point.
The subgroup of $GL(T_xM)$ generated by parallel 
translations (along all paths) is called {\bf \blue 
the holonomy group} of $M$.

{\bf\green REMARK:}
{\bf \purple A hyperk\"ahler manifold can be defined as a manifold which has
holonomy in $Sp(n)$ } (the group of all endomorphisms preserving
$I,J,K$).

\newpage


{\bf \blue Holomorphically symplectic manifolds}

{\bf\green DEFINITION:} A holomorphically symplectic manifold 
is a complex manifold equipped with non-degenerate, holomorphic
$(2,0)$-form.

\remark Hyperk\"ahler manifolds are holomorphically symplectic.
Indeed, $\Omega:=\omega_J+\1\omega_K$ is a holomorphic symplectic
form on $(M,I)$.

{\bf\green THEOREM:} (Calabi-Yau) 
A compact, K\"ahler, holomorphically symplectic manifold
admits a unique hyperk\"ahler metric in any K\"ahler class.

{\bf\green DEFINITION:}  For the rest of this talk, 
{\bf \red a hyperk\"ahler manifold
is a compact, K\"ahler, holomorphically symplectic manifold.}

{\bf\green DEFINITION:} A hyperk\"ahler manifold $M$ is called
{\bf \blue simple} if $\pi_1(M)=0$, $H^{2,0}(M)=\C$.

{\bf \purple Bogomolov's decomposition:} Any 
hyperk\"ahler manifold admits a finite covering
which is a product of a torus and several 
simple hyperk\"ahler manifolds.


{\bf \red Further on, all hyperk\"ahler manifolds
are assumed to be simple}.

\newpage

{\bf \blue The Bogomolov-Beauville-Fujiki form}

{\bf \green THEOREM:} (Fujiki). 
Let $\eta\in H^2(M)$, and $\dim M=2n$, where $M$ is
hyperk\"ahler. Then $\int_M \eta^{2n}=cq(\eta,\eta)^n$,
for some primitive integer quadratic form $q$ on $H^2(M,\Z)$,
and $c>0$ a rational number.

{\bf \green Definition:} This form is called
{\bf \blue Bogomolov-Beauville-Fujiki form}. {\bf \purple It is defined
by the Fujiki's relation uniquely, up to a sign.} The sign is determined
from the following formula (Bogomolov, Beauville)
\begin{align*}  \lambda q(\eta,\eta) &=
   \int_X \eta\wedge\eta  \wedge \Omega^{n-1}
   \wedge \bar \Omega^{n-1} -\\
 &-\frac {n-1}{n}\left(\int_X \eta \wedge \Omega^{n-1}\wedge \bar
   \Omega^{n}\right) \left(\int_X \eta \wedge \Omega^{n}\wedge \bar \Omega^{n-1}\right)
\end{align*}
where $\Omega$ is the holomorphic symplectic form, and 
$\lambda>0$.

{\bf \green Remark:}  {\bf \red $q$ has signature $(3,b_2-3)$.}
It is negative definite on primitive forms, and positive
definite on $\langle \Omega, \bar \Omega, \omega\rangle$,
 where $\omega$ is a K\"ahler form. 

\newpage

{\bf \blue Birational nef cone on a hyperk\"ahler manifold}

\definition
A cohomology class $\nu\in H^{1,1}(M)$ is called {\bf 
\blue pseudoeffective} if it can be represented by a positive,
closed current.



\theorem {\bf \blue (Huybrechts, Boucksom)}\\
On any hyperk\"ahler manifold $M$, {\bf \red birational K\"ahler cone
is dual (with respect to the BBF pairing) to the 
pseudoeffective cone.} Moreover, the birational K\"ahler cone
is a union of K\"ahler cones {\bf \red for all hyperk\"ahler 
manifolds $M'$ pseudo-isomorphic to $M$.}

\theorem {\bf \blue Divisorial 
Zariski decomposition (Boucksom). }\\
{\bf \red For any pseudoeffective $\nu$, we
have $\nu = \nu_0 + \sum \alpha_i E_i$,}
where $\nu_0$ is birationally nef, $\alpha_i$
positive numbers, and $E_i$ are exceptional divisors.

\corollary
Let $\eta\in \Pos(M)$ be an element of a positive cone
on a hyperk\"ahler manifold. {\bf \purple Then $\eta$ is birationally nef
if and only if $q(\eta, E)\geq 0$ for any exceptional divisor
$E$.}



\remark In other words, {\bf \red the faces of birational K\"ahler cone 
are dual to the classes of exceptional divisors.}


\newpage

{\bf \blue Divisorial reflections and Weyl chambers}


\definition
{\bf \blue Monodromy group} $\Gamma$ of a hyperk\"ahler manifold
is a subgroup of $O(H^2(M,\Z),q)$ generated by monodromy
operators for all Gauss-Manin local system associated with
complex deformations of $M$.

\remark The {\bf \purple monodromy group of a hyperk\"ahler manifold
is a finite index subgroup in $O(H^2(M,\Z),q)$} (follows
from global Torelli).

\theorem {\bf \blue (Markman)}\\
For each exceptional divisor $E$ on a hyperk\"ahler manifold,
{\bf \red there exists a reflection $r_E\in O(H^2(M,\Z))$ 
in the monodromy group fixing $E^\bot$.}

\definition
Such a reflection is called {\bf \blue a divisorial reflection}.

\definition
{\bf \blue Weyl chamber} on a hyperk\"ahler manifold 
is a connected component
of $\Pos(M)\backslash E^\bot$, where $E^{\bot}$ is a union of
all planes $e^\bot$ for all exceptional divisors $e$. 

\remark
A Weyl chamber is a fundamental domain of a group generated
by divisorial reflections. {\bf \purple Birational K\"ahler cone is one
of the Weyl chambers.}


\newpage

{\bf \blue The proof of birational cone conjecture}


\theorem
Let $M$ be a hyperk\"ahler manifold. 
{\bf \red Then $\Bir(M)$ is the group of
all $\gamma\in \Gamma$ preserving the birational K\"ahler chamber $\Kah_B$.}
Here $\Gamma$ is the monodromy group.\\
{\bf \green Proof:} Follows from global Torelli.
\endproof

\corollary {\bf \blue (Markman)} 
{\bf \red Birational 
Morrison-Kawamata cone conjecture holds for hyperk\"ahler manifolds.}

{\bf \green Proof. Step 1:} 
Let $\delta$ be the discriminant of a lattice $H^2(M,\Z)$,
and $E$ an exceptional divisor. {\bf \purple Then $|E^2|\leq 2\delta$.}
Indeed, otherwise the reflection $x \arrow x - 2\frac{q(x,E)}{q(E,E)}E$
would not be integer.

{\bf \green  Step 2:}  A group of isometries
of a lattice $\Lambda$ acts with finitely many orbits on the
set $\{l \in \Lambda\ \ |\ \ l^2 = x\}$
for any given $x$ (see Kneser, {\em Quadratische Formen},
Satz 30.2). Therefore, {\bf \purple $\Gamma$ acts with finitely many
orbits on the set of classes of exceptional divisors.}

{\bf \green Step 3:}
For each pair of faces $F, F'$ of a birational K\"ahler
cone and $w\in O(\Lambda)$ mapping $F$ to $F'$, $w$ maps $\Kah_B$ to itself
or to an adjoint Weyl chamber $K'$. Then $K'=r(K)$, where
$r$ is the reflection fixing $F'$.
In the first case, $w\in \Aut(M)$.
In the second case, $rw$ maps $F$ to $F'$ and maps $\Kah_B$ to itself,
hence $rw\in \Aut(M)$.
\endproof

\newpage

{\bf \blue MBM classes}

\definition
{\bf \blue Negative class} on a hyperk\"ahler manifold
is $\eta\in H^2(M,\R)$ satisfying $q(\eta,\eta)<0$.

\definition
Let $(M,I)$ be a hyperk\"ahler manifold.
A rational homology class
$z\in H_{1,1}(M,I)$ is called  {\bf \blue minimal}
if for any $\Q$-effective homology classes $z_1, z_2\in H_{1,1}(M,I)$
satisfying $z_1+z_2=z$, the classes $z_1, z_2$ are proportional.
A negative rational homology class
$z\in H_{1,1}(M,I)$ is called {\bf\blue monodromy birationally minimal} (MBM)
if $\gamma(z)$ is minimal and $\Q$-effective for one of birational
models $(M,I')$ of $(M,I)$, where $\gamma\in O(H^2(M))$
is an element of the monodromy group of $(M,I)$.

\definition
Let $(M,I)$ be a hyperkaehler manifold.
A negative rational class $z\in H^{1,1}_\Q(M,I)$ is called
{\bf\blue divisorial} if $z=\lambda [D]$ for some divisor $D$ and $\lambda\in \Q$.

These properties are {\bf \purple deformationally invariant}.

\theorem
Let $z\in H^2(M,\Z)$ be negative, and $I, I'$
complex structures in the same deformation class, such
that $\eta$ is of type (1,1) with respect to $I$ and $I'$.
Then \\
{\bf \red
{\ }\ \ \ * $\eta$ is divisorial in $(M,I)$ $\Leftrightarrow$ it is divisorial in $(M,I')$.\\
\bf \red {\ }\ \ \ * $\eta$ is MBM in $(M,I)$ $\Leftrightarrow$ it is MBM in $(M,I')$}.

\newpage

{\bf \blue MBM classes for $\Pic(M)=\Z$}

The MBM and divisorial classes are better understood if
the Picard group has rank one and generated by a negative vector
(in this case $M$ is non-algebraic).

\theorem
Let $(M,I)$ be a hyperk\"ahler manifold, $\rk \Pic(M,I)=1$, and
$z\in H_{1,1}(M,I)$ a non-zero negative class. {\bf \red Then $z$ is
monodromy birationally minimal if and only if 
$\pm z$ is $\Q$-effective.}

{\bf \green Proof. Step 1:} If $(M,I)$ has a rational curve, it is 
by definition minimal, hence represents an MBM class. 

{\bf \green Step 2:} If $(M,I)$ has no rational curves, it has
no exceptional divisors, hence {\bf \purple birational K\"ahler cone is
equal to the positive cone} (Boucksom).

{\bf \green Step 3:} If $(M,I)$ has no rational curves, any
pseudo-isomorphism from $(M,I)$ to another hyperk\"ahler manifold
must be trivial. Indeed, pseudo-isomorphisms are birational, and
exceptional locus of a birational map is covered by rational curves.
Then $\Kah_B=\Kah$: {\bf \red birational K\"ahler cone is K\"ahler cone 
is positive cone.}
\endproof

\remark 
This argument proves that 
{\bf \purple MBM classes correspond to faces of a K\"ahler cone}
for $\rk \Pic(M,I)=1$.

\newpage

{\bf \blue MBM classes and the K\"ahler cone}

\theorem
Let $(M,I)$ be a hyperk\"ahler manifold, 
and $S\subset H_{1,1}(M,I)$ the set of all MBM classes in 
$H_{1,1}(M,I)$. Consider the corresponding set of hyperplanes
$S^\bot:=\{W=z^\bot\ \ |\ \ z\in S\}$ in $H^{1,1}(M,I)$.
{\bf \red Then the K\"ahler cone of $(M,I)$ is 
a connected component of $\Pos(M,I)\backslash \cup S^\bot$},
where $\Pos(M,I)$ is a positive cone of $(M,I)$.
Moreover, for any connected component $K$ of $\Pos(M,I)\backslash \cup S^\bot$,
there exists $\gamma\in O(H^2(M))$ in a monodromy group of $M$, and
a hyperk\"ahler manifold $(M,I')$ birationally equivalent 
to $(M,I)$, such that $\gamma(K)$ is a K\"ahler cone of $(M,I')$.

\remark
{\bf \purple MBM classes correspond to faces of the K\"ahler cone.}

\remark {\bf \red Morrison-Kawamata cone conjecture would
follow if we prove that monodromy group acts on
the set of MBM rays with finitely many orbits.}
This is implied by the following conjecture.

\conjecture Let $M$ be a hyperk\"ahler manifold.
{\bf \purple Then there exists a constant $C>0$, such that for
any minimal rational curve $S\subset (M,I)$ with $q(S,S)<0$, 
one has $|q(S,S)|<C$.}


\end{document}



